Theorem pweqb 4208

Description: Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)

Assertion

Ref	Expression
pweqb	⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)

Proof of Theorem pweqb

Step	Hyp	Ref	Expression
1		sspwb 4201	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
2		sspwb 4201	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴)
3	1, 2	anbi12i 457	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴))
4		eqss 3162	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
5		eqss 3162	. 2 ⊢ (𝒫 𝐴 = 𝒫 𝐵 ↔ (𝒫 𝐴 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴))
6	3, 4, 5	3bitr4i 211	1 ⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1348   ⊆ wss 3121  𝒫 cpw 3566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589

This theorem is referenced by: (None)